Problem: Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{5}{4}\pi) + i \sin(\frac{5}{4}\pi)}) ^ {7} $
Answer: Let's express our complex number in Euler form first. $ {\cos(\frac{5}{4}\pi) + i \sin(\frac{5}{4}\pi)} = { e^{5\pi i / 4}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{5\pi i / 4}}) ^ {7} = e ^ {7 \cdot (5\pi i / 4)} $ The angle of the result is $7 \cdot \frac{5}{4}\pi$ , which is $\frac{35}{4}\pi$ $\frac{35}{4}\pi$ is more than $2 \pi$ . It is a common practice to keep complex number angles between $0$ and $2 \pi$ , because $e^{2 \pi i} = (e^{\pi i}) ^ 2 = (-1) ^ 2 = 1$ . We will now subtract the nearest multiple of $2 \pi$ from the angle. $ \frac{35}{4}\pi - 8\pi = \frac{3}{4}\pi $ Our result is $ e^{3\pi i / 4}$. Converting this back from Euler form, we get $\cos(\frac{3}{4}\pi) + i \sin(\frac{3}{4}\pi)$.